Post on 30-Mar-2015
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BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx1
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering 36
Chp 7:Flex
Cables
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx2
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Recall Chp10 Introduction
Examine in Detail Two Important Types Of Engineering Structures:
1. BEAMS - usually long, straight, prismatic members designed to support loads applied at various points along the member
2. CABLES - flexible members capable of withstanding only tension, designed to support concentrated or distributed loads
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx3
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Load-Bearing Cables
Cables are applied as structural elements in suspension bridges, transmission lines, aerial tramways, guy wires for high towers, etc.
Göteborg, Sweden
http://www.nmt.edu/~armiller/bridgefu.htm
CurvedStraight
WHY the Difference?
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx4
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Concentrated Loads on Cables To Determine the Cable
SHAPE, Assume:a) Concentrated vertical loads
on given vertical lines
b) Weight of cable is negligible
c) Cable is flexible, i.e., resistance to bending is small
d) Portions of cable between successive loads may be treated as TWO FORCE MEMBERS• Internal Forces at any point reduce to
TENSION Directed ALONG the Cable Axis
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx5
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Concentrated Loads (2) Consider entire cable
as a free-body Slopes of cable at A
and B are NOT known FOUR unknowns
(i.e., Ax, Ay, Bx, By) are involved and the equations of equilibrium are NOT sufficient to determine the reactions.
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx6
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Concentrated Loads (3) To Obtain an additional equation • Consider equilibrium of
cable-section AD• Assume that CoOrdinates of SOME
point D, (x,y), on the cable have been Determined (e.g., by measurement)
Then the added Eqn: 0DM
33221103 xPxPxPLBdBM yxA
With pt-D info, the FOUR Equilibrium Eqns xxx BAF 01̀ 12102 PPPBAF yyy
xAyAxxPM yxD 1104
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx7
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Concentrated Loads (4) The 4 Eqns Yield Ax & Ay
Can Now Work our Way Around the Cable to Find VERTICAL DISTANCE (y-CoOrd) For ANY OTHER point
2 yields02
yMC yxyx TTFF , yields 0,0
Example Consider Pt C2
constantcos xx ATT
known
known
known
UNknown
known
knownknown
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx8
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example Concentrated Loads
The cable AE supports three vertical loads from the points indicated. Point C is 5 ft below the left support
For the Given Loading & Geometry, Determine:a) The elevation of
points B and D b) The maximum
slope and maximum tension in the cable.
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx9
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example Concentrated Loads
Solution Plan• Determine
reaction force components at pt-A from solution of two equations formed from taking entire cable as a
free-body and summing moments about E, and from taking cable portion ABC as a free-body and summing moments about C.
• Calculate elevation of B by considering AB as a free-body and summing moments B. Similarly, calculate elevation of D using ABCD as a free-body.
• Evaluate maximum slope and maximum tension which occur in DE.
KnownCoOrds
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx10
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example Concentrated Loads
Determine two reaction force components at A from solution of two equations formed from taking the
entire cable as a free-body and summing moments about E:
333
OR
06606020
041512306406020
:0
yx
yx
yx
E
AA
AA
AA
M
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx11
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example Concentrated Loads
Next take Cable Section ABC as a Free-Body, and Sum the Moments about Point C
Recall from ΣME
126OR
0610305
:0
yx
yx
C
AA
AA
M
333 yx AA Solving 2-Eqns in
2-Unknowns for Ax & Ay
kips 5
kips 18
y
x
A
A
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx12
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example Concentrated Loads Determine elevation of B
by considering AB as a free-body and summing moments about B.
Similarly, Calc elevation at D using ABCD as a free-body
020518:0 BB yM
ft 56.5By
0121562554518
:0
D
D
y
M
ft83.5Dy
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx13
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Maximum Tension Analysis
By the ΣFx = 0
T
cos
but
OR0
TT
AT
TAF
x
xx
xxx
Solving
for T cosxAT
Thus T is Maximized by Maximum θ
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx14
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Find Maximum Segment Angle
Use the y-data just calculated to find the cable segment of steepest slope
Pt x (ft) y (ft) Seg ∆x (ft) ∆y (ft) θ by aTan(∆y/∆x) (°)A 0 0
AB 20 -5.56 -15.54B 20 -5.56
BC 10 0.56 3.21C 30 -5
CD 15 10.83 35.83D 45 5.83
DE 15 14.17 43.37E 60 20
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx15
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example Concentrated Loads
Use yD to Determine Geometry of tanθ
Evaluate maximum slope and maximum tension which occur in the segment with the STEEPEST Slope (large θ); DE in this case
15
17.14tan 4.43
Employ the Just-Determined θ to Find Tmax
cos
kips 18max T
kips 8.24max T
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx16
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads on Cables For a negligible-
Weight Cable carrying a Distributed Load ofArbitrary Profile
a) The cable hangs in shape of a CURVE
b) INTERNAL force is a tension force directed along the TANGENT to the curve
x
y
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx17
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads (2) Consider the Free-Body for a
portion of cable extending from LOWEST point C to given point D. Forces are T0 (FH in the Text Book) at Lo-Pt C, and the tangential force T at D
From the Force Triangle
0
220
0
tan
sincos
T
WWTT
WTTT
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx18
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads (3) Some Observations based on
0
220
0
tan
sincos
T
WWTT
WTTTTT yx
• Horizontal component of T is uniform over the Cable Length
• Vertical component of T is equal to the magnitude of W
• Tension is minimum at lowest point (min θ), and maximum at A and B (max θ)
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx19
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Parabolic Cable Consider a cable supporting a
uniform, horizontally distributed load, e.g., support cables for a suspension bridge.
With loading on cable from lowestpoint C to a point D given by internal tension force, T, and Vertical Load W = wx, find:
02
:0 0 yTx
wxMD
0
2
2T
wxy
0
2220 tan
T
wxxwTT
Summing moments about D
The shape, y, is PARABOLIC:
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx20
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
T0 for Uniform Vertical Load
Consider the uniformly Loaded Cable• In this case: w(x) = w
– w is a constant
L is the Suspension Span:
From Last Slide
Or Then
xB & xA
Thus L
L
AB xxL
0
2
2T
wxy
wyTx 02
wyTx
wyTx
AA
BB
0
0
2
2
wyTwyTL AB 00 22
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx21
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
T0 for Uniform Vertical Load
Factoring Out 2T0/w
Isolating T0
If WE design the Suspension System, then we KNOW• L (Span)• w (Load)• yA & yB (Dims)
Example• L= 95 m (312 ft)• w = 640 N/m
(44 lb/ft)• yA = 19m
• yB = 37m
Then T0 = 26 489 N (5955 lb)
Tmax by
AB yywTL 20
22
02 AB yy
wLT
2max
220max xwTT
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx22
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Tmax for Uniform Vertical Load
Find xmax from
In this case xmax = 55.3 m (181 ft)
And finally Tmax = 44 228 N(9 943 lbs)• Buy Cable rated to
20 kip for Safety factor of 2.0
wyTx BB 02
>> L = 95L = 95 >> w = 640w = 640 >> yA = 19yA = 19 >> yB = 37yB = 37 >> TO =L^2*w/(2*(sqrt(yB)+sqrt(yA))^2)TO = 2.6489e+04 >> xB = sqrt(2*TO*yB/w)xB = 55.3420 >> Tmax = sqrt(TO^2 + (w*xB)^2)Tmax = 4.4228e+04
MATLABCalcs
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx23
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Load-Bearing Cables
Tension in the Straight-Section is roughly Equal to the Parabolic-Tension at the Tower-Top.• So the Support Tower does Not Bend
Göteborg, Sweden
http://www.nmt.edu/~armiller/bridgefu.htm
VerticallyLoaded
End Loads
NO Deck-Support Cables
Deck-Support COLUMNS
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx24
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary Consider a cable uniformly
loaded by the cable itself, e.g., a cable hanging under its own weight.
222220
22220 scwswTwswTT
• With loading on the cable from lowest point C to a point D given by W = ws, the Force Triangle reveals the internal tension force magnitude at Pt-D:
mor ft 0 wTc – Where
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx25
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary (2) Next, relate horizontal
distance, x, to cable-length s
cosdsdx But by Force
Balance Triangle
Thus
T
T0cos
Also From last slide recall
wcTscwT 022 and
dssc
c
scw
wcds
T
Tdsdsdx
2222
0cos
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx26
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary (3) Factoring Out c in DeNom
Integrate Both Sides using Dummy Variables of Integration: • σ: 0→x η: 0→s
dscsccc
cds
sc
cdx
222222
Finally the Differential Eqn
dscs
dx221
1
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx27
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary (4) Using σ: 0→x η: 0→s
Now the R.H.S. AntiDerivative is the argSINH
Noting that
sxd
cd
0 220 1
1
s
sxx
ccd
cd
0
0 2200sinharg
1
1
00sinh0sinharg 1
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx28
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary (5) Thus the Solution to the Integral Eqn
Then
Solving for s in terms of x by taking the sinh of both sides
0sinhsinharg0 1
0
0
c
sc
ccx
sx
c
s
c
x
c
scx 11 sinhsinh
c
x
c
ssinh
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx29
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary (6) Finally, Eliminate s in favor
of x & y. From the Diagram
So the Differential Eqn
From the Force Triangle
tandxdy
0
tanT
W
And From Before
wcTwsW 0and
dxc
sdx
wc
wsdx
T
Wdxdy
0
tan
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx30
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary (7) Recall the Previous Integration
That Relates x and s
Integrating the ODE with Dummy Variables: • Ω: c→y σ: 0→x
x
xyc
y
c ccd
cd
00
coshsinh
c
x
c
ssinh
Using s(x) above in the last ODE
dxc
xdxc
sdx
wc
wsdx
T
Wdxdy
o
sinhtan
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx31
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
UNloaded Cable → Catenary (8) Noting that cosh(0) = 1
• Where– c = T0/w
– T0 = the 100% laterally directed force at the ymin point
– w = the lineal unit weight of the cable (lb/ft or N/m)
Solving for y yields theCatenary Equation:
cxcy cosh
cc
xc
cccy
xyc
coshcosh
0
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx32
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Catenary Comments
With Hyperbolic-Trig ID: cosh2 – sinh2 = 1
Recall From the Differential Geometry
Or:
cxcy cosh
222222
222222
sinhcosh
sinhcosh
ccxcxcsy
cxccxcsy
222222 sycandscy
yTwyywscwscT 222,
wTc 0
or
00
0
0 coshcoshT
wxT
T
wx
w
TwwyxT
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx33
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Loaded & Unloaded Cable Profiles
0
50
100
150
200
250
300
350
400
450
500
550
-100 -75 -50 -25 0 25 50 75 100
Relative Horizontal Displacement, x
Rel
ativ
e V
erti
cal
Dis
pla
cem
ent,
y
Parabola (Loaded)
Catenary (Unloaded)
file = Catenary-Parabola_0407.xls
PARAMETERS• w/T0 = 0.1 (Parabola)• c = 50 (Catenary)
Loaded and Unloaded Cables Compared
cxcy cosh
0
2
2T
wxy
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx34
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
y = 0 at Cable Minimum
Translate the CoOrd System Vertically from Previous:
Oy
cyycyy
xx
OO Change No
Recall Eqn for y−c
cc
xccy
cosh
Sub y = yO+c
Thus with Origin at cable Minimum
cc
xcccyO
cosh
cc
xcyO
cosh
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx35
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
y = 0 at Cable Minimum (2)
ThenOy
Recall c = T0/w
Thus Next, Change the Name of the Cable’s Lineal Specific Weight (N/m or lb/ft)
1c
xcyO cosh
1
0
0
wT
x
w
TyO cosh
w
L
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx36
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
y = 0 at Cable Minimum (3)
With µ replacing w
Oy
1
0
0
T
xTyO
cosh
In Summary can Use either Formulation based on Axes Origin:
0
0 coshT
wx
w
Ty
1
0
0
T
xTyO
cosh
wT0
L
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx37
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Cable Length, S, for Catenary Using this Axes Set
With Cable macro-segment and differential-segment at upper-right
The Force Triangle for the Macro-Segment
OyL
W = µs
W = µs
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx38
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Cable Length, S , for Catenary By Force
Triangle But by Differential
Segment notice
By Transitive Property
Now ReCall
Then dy/dx
Subbing into the tanθ expression
0
tanT
s
dx
dytan
0
tanT
µs
dx
dy
1cosh
0
0
T
xTy
00
0 sinh1coshT
x
T
xT
dx
d
dx
dy
00
sinhT
µs
T
µx
dx
dy
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx39
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Cable Length, S , for Catenary Solving the last Eqn
for s
From the CoOrds
So Finally
Now find T(y) Recall T = f(x)
0
0 sinhT
µx
µ
Ts
OyL
AB
OAOB
xsxs
ssS
00
0 sinhsinhT
µx
T
µx
µ
TS AB
00 cosh
T
xTT
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx40
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
T(y) for Catenary Also ReCall
Solve above for cosh
Sub cosh into T(x) Expression
1cosh
0
0
T
xTy
11cosh000
T
µy
T
µy
T
x
00
0
00
1
cosh
TµyT
µyTT
T
µxTT
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx41
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Catenary Summary y(x)
T(x)T(y)
S(x)
Slope at any pt
Angle θ at any pt
1cosh
0
0
T
xTy
0
00 cosh
TµyT
T
µxTT
00
0 sinhsinhT
µx
T
µx
µ
TS AB
00
sinhT
µs
T
µx
dx
dy
00
sinhtanT
µs
T
µx
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx42
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
WhiteBoard Work
Let’s WorkThese NiceProblems
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx43
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PERegistered Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering 36
Appendix00
sinhT
µs
T
µx
dx
dy
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx44
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
WhiteBoard Work
Let’s WorkThese NiceProblems
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx45
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
w
T0
BMayer@ChabotCollege.edu • ENGR-36_Lec-20_Cables.pptx46
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics